\newproblem{lay:6_7_16}{
  % Problem identification
	\begin{large}
	  \hspace{\fill}\newline
    \textbf{Lay, 6.7.16}
	\end{large}
	\\
  \ifthenelse{\boolean{identifyAuthor}}{\textit{Carlos Oscar Sorzano, Aug. 31st, 2013} \\}{}

  % Problem statement
	Show that if $S=\{\mathbf{u},\mathbf{v}\}$ is an orthonormal set in $V$, then $\|\mathbf{u}-\mathbf{v}\|=\sqrt{2}$.
}{
   % Solution
	If $S$ is orthonormal, then $\|\mathbf{u}\|=\|\mathbf{v}\|=1$ and $\mathbf{u}\cdot\mathbf{v}=0$. Then,
	\begin{center}
		$\begin{array}{rcl}
			\|\mathbf{u}-\mathbf{v}\| &=&\sqrt{(\mathbf{u}-\mathbf{v})\cdot(\mathbf{u}-\mathbf{v})} \\
			  &=& \sqrt{\mathbf{u}\cdot\mathbf{u}+\mathbf{v}\cdot\mathbf{v}-2\mathbf{u}\cdot\mathbf{v}} \\
			  &=& \sqrt{\|\mathbf{u}\|^2+\|\mathbf{v}\|^2-2\mathbf{u}\cdot\mathbf{v}} \\
			  &=& \sqrt{1+1-2\cdot 0} \\
			  &=& \sqrt{2} \\
		\end{array}$
	\end{center}
}
\useproblem{lay:6_7_16}
\ifthenelse{\boolean{eachProblemInOnePage}}{\newpage}{}

